From Bucket-Elimination To Bucket Trees E Bucket E: P(E|B,C) - PDF document

From Bucket-Elimination To Bucket Trees E Bucket E: P(E|B,C) P(E|B,C) D λ Bucket D: P(D|A,B) P(D|A,B) B C ( , ) C λ Bucket C: P(C|A) P(C|A) λ C A B ( , ) B C ( , ) B E λ ( A , B ) Bucket B: P(B|A) λ λ P(B|A) B B A B ( A , B ) ( , ) A D C λ Bucket A: P(A) P(A) λ ( A ) ( A ) B B E E B,C D D D,B,A C C T = A,B,C B B B,A A A A Definition: T is a bucket tree. Theorem: T is an i-map of G. • Variable-elimination can be viewed as message-passing (elimination) using a rooted bucket tree. • Any variable (bucket) can be the root.

Generalization:Eliminate (sum over) Variables Not in Separators B 3 B 4 λ 1 λ 2 B 1 P i S=B 1 ∩ B 2 S B 2 • Multiply all incoming messages, and P i ’s in the bucket and sum over B 1 ∩ B 2 . λ = Π λ Π ∑ B • 2 s P ( ) ( ) ( ) • i i B 1 − B s 1 • Given a rooted bucket tree, T, every node can be the “root” of the variables-elimination computation. • If B 3 is the root, bucket B 2 and then Bucket B 1 should be processed; π -messages sent from B 2 to B 1 and from B 1 To B 3

Bucket Propagation Algorithm • Input: A bucket tree B 1 …B n • Output: For Each B i and parent B j , functions λ i j (S ij ) and π i j (S ij ) are exchanged. B k λ 1 Π B i λ ∩ = B B S j ij i B j Top Down: • Let s λ 1 … λ k messages from child nodes of B i , P 1 …P r in B i original functions. = ∑ λ Π λ • Π j S P • ( ) i ij i j i j − B B i j Bottom Up: • Let π i j be received from B j. ∑ = Π • π • λ π i Π k S P ( ) ( ) • i ki j j i j ≠ i k − B B k i

• The belief of B i Π • Π λ • π • P ( B i ) = i i P j j i i • if x index Bucket i ∑ get Bel( x ) by summing out Bel( x ) = α P B ( ) i S ij

Propagation in a Bucket Tree Definitions: • Let G be a Bayesian network, d , an ordering and B 1 … B n the final bucket created processing along d = x 1 … x n . • Let B i be the set of variables appearing in bucket i when it is processed. Bucket Tree: • A bucket tree has each B i cluster as a node and there is an arc from B i to B j if the function created at B i was placed in B j Graph-Based Definition: • Let G d be the induced graph along d . Each variable x and it’s earlier neighbors in a node, B x . There is an arc from B x to B y if y is the closest parent of x.

Upwards Messages On The Bucket Tree E E,B,C D λ B C ( , ) A,B,D C λ A B ( , ) λ A B ( , ) A,B,C B Π A B ( , ) B,A B A Π λ ( A A ) Π = A P A ( ) ( ) Π = • λ P B A B P B A A B ( , ) ( , ) ( , ) B C Π = • Π • λ C B A B P B A A A B ( , ) ( , ) ( ) ( , ) B D Π = • Π ∑ E C B C P C A A B ( , ) ( , ) ( , ) C B A